Detailed Description

A Gaussian factor using the canonical parameters (information form)

HessianFactor implements a general quadratic factor of the form

\[ E(x) = 0.5 x^T G x - x^T g + 0.5 f \]

that stores the matrix \( G \), the vector \( g \), and the constant term \( f \).

When \( G \) is positive semidefinite, this factor represents a Gaussian, in which case \( G \) is the information matrix \( \Lambda \), \( g \) is the information vector \( \eta \), and \( f \) is the residual sum-square-error at the mean, when \( x = \mu \).

Indeed, the negative log-likelihood of a Gaussian is (up to a constant) \( E(x) = 0.5(x-\mu)^T P^{-1} (x-\mu) \) with \( \mu \) the mean and \( P \) the covariance matrix. Expanding the product we get

\[ E(x) = 0.5 x^T P^{-1} x - x^T P^{-1} \mu + 0.5 \mu^T P^{-1} \mu \]

We define the Information matrix (or Hessian) \( \Lambda = P^{-1} \) and the information vector \( \eta = P^{-1} \mu = \Lambda \mu \) to arrive at the canonical form of the Gaussian:

\[ E(x) = 0.5 x^T \Lambda x - x^T \eta + 0.5 \mu^T \Lambda \mu \]

This factor is one of the factors that can be in a GaussianFactorGraph. It may be returned from NonlinearFactor::linearize(), but is also used internally to store the Hessian during Cholesky elimination.

This can represent a quadratic factor with characteristics that cannot be represented using a JacobianFactor (which has the form \( E(x) = \Vert Ax - b \Vert^2 \) and stores the Jacobian \( A \) and error vector \( b \), i.e. is a sum-of-squares factor). For example, a HessianFactor need not be positive semidefinite, it can be indefinite or even negative semidefinite.

If a HessianFactor is indefinite or negative semi-definite, then in order for solving the linear system to be possible, the Hessian of the full system must be positive definite (i.e. when all small Hessians are combined, the result must be positive definite). If this is not the case, an error will occur during elimination.

This class stores G, g, and f as an augmented matrix HessianFactor::matrix_. The upper-left n x n blocks of HessianFactor::matrix_ store the upper-right triangle of G, the upper-right-most column of length n of HessianFactor::matrix_ stores g, and the lower-right entry of HessianFactor::matrix_ stores f, i.e.

HessianFactor::matrix_ = [ G11 G12 G13 ... g1
                             0 G22 G23 ... g2
                             0   0 G33 ... g3
                             :   :   :      :
                             0   0   0 ...  f ]

Blocks can be accessed as follows:

G11 = info(begin(), begin());
G12 = info(begin(), begin()+1);
G23 = info(begin()+1, begin()+2);
g2 = linearTerm(begin()+1);
f = constantTerm();
.......

Inheritance diagram for gtsam::HessianFactor:

Public Member Functions
	HessianFactor ()
	default constructor for I/O

	HessianFactor (Key j, const Matrix &G, const Vector &g, double f)
	Construct a unary factor. More...

	HessianFactor (Key j, const Vector &mu, const Matrix &Sigma)
	Construct a unary factor, given a mean and covariance matrix. More...

	HessianFactor (Key j1, Key j2, const Matrix &G11, const Matrix &G12, const Vector &g1, const Matrix &G22, const Vector &g2, double f)
	Construct a binary factor. More...

	HessianFactor (Key j1, Key j2, Key j3, const Matrix &G11, const Matrix &G12, const Matrix &G13, const Vector &g1, const Matrix &G22, const Matrix &G23, const Vector &g2, const Matrix &G33, const Vector &g3, double f)
	Construct a ternary factor. More...

	HessianFactor (const std::vector< Key > &js, const std::vector< Matrix > &Gs, const std::vector< Vector > &gs, double f)
	Construct an n-way factor. More...

template<typename KEYS >
	HessianFactor (const KEYS &keys, const SymmetricBlockMatrix &augmentedInformation)
	Constructor with an arbitrary number of keys and with the augmented information matrix specified as a block matrix. More...

	HessianFactor (const JacobianFactor &cg)
	Construct from a JacobianFactor (or from a GaussianConditional since it derives from it)

	HessianFactor (const GaussianFactor &factor)
	Attempt to construct from any GaussianFactor - currently supports JacobianFactor, HessianFactor, GaussianConditional, or any derived classes. More...

	HessianFactor (const GaussianFactorGraph &factors, boost::optional< const Scatter & > scatter=boost::none)
	Combine a set of factors into a single dense HessianFactor.

virtual	~HessianFactor ()
	Destructor.

virtual GaussianFactor::shared_ptr	clone () const
	Clone this HessianFactor.

virtual void	print (const std::string &s="", const KeyFormatter &formatter=DefaultKeyFormatter) const
	Print the factor for debugging and testing (implementing Testable)

virtual bool	equals (const GaussianFactor &lf, double tol=1e-9) const
	Compare to another factor for testing (implementing Testable)

virtual double	error (const VectorValues &c) const
	Evaluate the factor error f(x), see above. More...

virtual DenseIndex	getDim (const_iterator variable) const
	0.5*[x -1]'H[x -1] (also see constructor documentation) More...

size_t	rows () const
	Return the number of columns and rows of the Hessian matrix, including the information vector. More...

virtual GaussianFactor::shared_ptr	negate () const
	Construct the corresponding anti-factor to negate information stored stored in this factor. More...

virtual bool	empty () const
	Check if the factor is empty. More...

constBlock	info (const_iterator j1, const_iterator j2) const
	Return a view of the block at (j1,j2) of the upper-triangular part of the information matrix \( H \), no data is copied. More...

Block	info (iterator j1, iterator j2)
	Return a view of the block at (j1,j2) of the upper-triangular part of the information matrix \( H \), no data is copied. More...

SymmetricBlockMatrix::constBlock	info () const
	Return the upper-triangular part of the full augmented information matrix, as described above. More...

SymmetricBlockMatrix::Block	info ()
	Return the upper-triangular part of the full augmented information matrix, as described above. More...

double	constantTerm () const
	Return the constant term \( f \) as described above. More...

double &	constantTerm ()
	Return the constant term \( f \) as described above. More...

constBlock::OffDiagonal::ColXpr	linearTerm (const_iterator j) const
	Return the part of linear term \( g \) as described above corresponding to the requested variable. More...

Block::OffDiagonal::ColXpr	linearTerm (iterator j)
	Return the part of linear term \( g \) as described above corresponding to the requested variable. More...

constBlock::OffDiagonal::ColXpr	linearTerm () const
	Return the complete linear term \( g \) as described above. More...

Block::OffDiagonal::ColXpr	linearTerm ()
	Return the complete linear term \( g \) as described above. More...

virtual Matrix	augmentedInformation () const
	Return the augmented information matrix represented by this GaussianFactor. More...

virtual Matrix	information () const
	Return the non-augmented information matrix represented by this GaussianFactor.

virtual VectorValues	hessianDiagonal () const
	Return the diagonal of the Hessian for this factor.

virtual void	hessianDiagonal (double *d) const
	Return the diagonal of the Hessian for this factor (raw memory version)

virtual std::map< Key, Matrix >	hessianBlockDiagonal () const
	Return the block diagonal of the Hessian for this factor.

virtual std::pair< Matrix, Vector >	jacobian () const
	Return (dense) matrix associated with factor. More...

virtual Matrix	augmentedJacobian () const
	Return (dense) matrix associated with factor The returned system is an augmented matrix: [A b]. More...

const SymmetricBlockMatrix &	matrixObject () const
	Return the full augmented Hessian matrix of this factor as a SymmetricBlockMatrix object. More...

void	updateATA (const JacobianFactor &update, const Scatter &scatter)
	Update the factor by adding the information from the JacobianFactor (used internally during elimination). More...

void	updateATA (const HessianFactor &update, const Scatter &scatter)
	Update the factor by adding the information from the HessianFactor (used internally during elimination). More...

void	multiplyHessianAdd (double alpha, const VectorValues &x, VectorValues &y) const
	y += alpha * A'Ax

void	multiplyHessianAdd (double alpha, const double x, double y, std::vector< size_t > keys) const
	y += alpha * A'Ax

void	multiplyHessianAdd (double alpha, const double x, double y) const
	y += alpha * A'Ax

VectorValues	gradientAtZero () const
	eta for Hessian

virtual void	gradientAtZero (double *d) const
	A'*b for Jacobian, eta for Hessian (raw memory version)

Public Member Functions inherited from gtsam::GaussianFactor
	GaussianFactor ()
	Default constructor creates empty factor.

template<typename CONTAINER >
	GaussianFactor (const CONTAINER &keys)
	Construct from container of keys. More...

virtual	~GaussianFactor ()
	Destructor.

Public Member Functions inherited from gtsam::Factor
Key	front () const
	First key.

Key	back () const
	Last key.

const_iterator	find (Key key) const
	find

const FastVector< Key > &	keys () const
	Access the factor's involved variable keys.

const_iterator	begin () const
	Iterator at beginning of involved variable keys.

const_iterator	end () const
	Iterator at end of involved variable keys.

size_t	size () const

void	print (const std::string &s="Factor", const KeyFormatter &formatter=DefaultKeyFormatter) const
	print

void	printKeys (const std::string &s="Factor", const KeyFormatter &formatter=DefaultKeyFormatter) const
	print only keys

FastVector< Key > &	keys ()

iterator	begin ()
	Iterator at beginning of involved variable keys.

iterator	end ()
	Iterator at end of involved variable keys.

Public Types
typedef GaussianFactor	Base
	Typedef to base class.

typedef HessianFactor	This
	Typedef to this class.

typedef boost::shared_ptr< This >	shared_ptr
	A shared_ptr to this class.

typedef SymmetricBlockMatrix::Block	Block
	A block from the Hessian matrix.

typedef SymmetricBlockMatrix::constBlock	constBlock
	A block from the Hessian matrix (const version)

Public Types inherited from gtsam::GaussianFactor
typedef GaussianFactor	This
	This class.

typedef boost::shared_ptr< This >	shared_ptr
	shared_ptr to this class

typedef Factor	Base
	Our base class.

Public Types inherited from gtsam::Factor
typedef FastVector< Key >::iterator	iterator
	Iterator over keys.

typedef FastVector< Key > ::const_iterator	const_iterator
	Const iterator over keys.

Protected Attributes
SymmetricBlockMatrix	info_
	The full augmented information matrix, s.t. the quadratic error is 0.5*[x -1]'H[x -1].

Protected Attributes inherited from gtsam::Factor
FastVector< Key >	keys_
	The keys involved in this factor.

Friends
class	boost::serialization::access
	Serialization function.

GTSAM_EXPORT std::pair < boost::shared_ptr < GaussianConditional > , boost::shared_ptr < HessianFactor > >	EliminateCholesky (const GaussianFactorGraph &factors, const Ordering &keys)

GTSAM_EXPORT std::pair < boost::shared_ptr < GaussianConditional > , boost::shared_ptr < GaussianFactor > >	EliminatePreferCholesky (const GaussianFactorGraph &factors, const Ordering &keys)

Constructor & Destructor Documentation

gtsam::HessianFactor::HessianFactor	(	Key	j,
		const Matrix &	G,
		const Vector &	g,
		double	f
	)

Construct a unary factor.

G is the quadratic term (Hessian matrix), g the linear term (a vector), and f the constant term. The quadratic error is: 0.5*(f - 2*x'*g + x'*G*x)

gtsam::HessianFactor::HessianFactor	(	Key	j,
		const Vector &	mu,
		const Matrix &	Sigma
	)

Construct a unary factor, given a mean and covariance matrix.

error is 0.5*(x-mu)'inv(Sigma)(x-mu)

gtsam::HessianFactor::HessianFactor	(	Key	j1,
		Key	j2,
		const Matrix &	G11,
		const Matrix &	G12,
		const Vector &	g1,
		const Matrix &	G22,
		const Vector &	g2,
		double	f
	)

Construct a binary factor.

Gxx are the upper-triangle blocks of the quadratic term (the Hessian matrix), gx the pieces of the linear vector term, and f the constant term. JacobianFactor error is

\[ 0.5* (Ax-b)' M (Ax-b) = 0.5*x'A'MAx - x'A'Mb + 0.5*b'Mb \]

HessianFactor error is

\[ 0.5*(x'Gx - 2x'g + f) = 0.5*x'Gx - x'*g + 0.5*f \]

So, with \( A = [A1 A2] \) and \( G=A*'M*A = [A1';A2']*M*[A1 A2] \) we have

n1*n1 G11 = A1'*M*A1
n1*n2 G12 = A1'*M*A2
n2*n2 G22 = A2'*M*A2
n1*1   g1 = A1'*M*b
n2*1   g2 = A2'*M*b
 1*1    f =  b'*M*b

gtsam::HessianFactor::HessianFactor	(	Key	j1,
		Key	j2,
		Key	j3,
		const Matrix &	G11,
		const Matrix &	G12,
		const Matrix &	G13,
		const Vector &	g1,
		const Matrix &	G22,
		const Matrix &	G23,
		const Vector &	g2,
		const Matrix &	G33,
		const Vector &	g3,
		double	f
	)

Construct a ternary factor.

Gxx are the upper-triangle blocks of the quadratic term (the Hessian matrix), gx the pieces of the linear vector term, and f the constant term.

gtsam::HessianFactor::HessianFactor	(	const std::vector< Key > &	js,
		const std::vector< Matrix > &	Gs,
		const std::vector< Vector > &	gs,
		double	f
	)

Construct an n-way factor.

Gs contains the upper-triangle blocks of the quadratic term (the Hessian matrix) provided in row-order, gs the pieces of the linear vector term, and f the constant term.

template<typename KEYS >

gtsam::HessianFactor::HessianFactor	(	const KEYS &	keys,
		const SymmetricBlockMatrix &	augmentedInformation
	)

Constructor with an arbitrary number of keys and with the augmented information matrix specified as a block matrix.

gtsam::HessianFactor::HessianFactor ( const GaussianFactor & factor )

explicit

Attempt to construct from any GaussianFactor - currently supports JacobianFactor, HessianFactor, GaussianConditional, or any derived classes.

Member Function Documentation

Matrix gtsam::HessianFactor::augmentedInformation ( ) const

virtual

Return the augmented information matrix represented by this GaussianFactor.

The augmented information matrix contains the information matrix with an additional column holding the information vector, and an additional row holding the transpose of the information vector. The lower-right entry contains the constant error term (when \( \delta x = 0 \)). The augmented information matrix is described in more detail in HessianFactor, which in fact stores an augmented information matrix.

For HessianFactor, this is the same as info() except that this function returns a complete symmetric matrix whereas info() returns a matrix where only the upper triangle is valid, but should be interpreted as symmetric. This is because info() returns only a reference to the internal representation of the augmented information matrix, which stores only the upper triangle.

Implements gtsam::GaussianFactor.

Matrix gtsam::HessianFactor::augmentedJacobian ( ) const

virtual

Return (dense) matrix associated with factor The returned system is an augmented matrix: [A b].

Parameters

set	weight to use whitening to bake in weights

Implements gtsam::GaussianFactor.

double gtsam::HessianFactor::constantTerm ( ) const

inline

Return the constant term \( f \) as described above.

Returns: The constant term \( f \)

double& gtsam::HessianFactor::constantTerm ( )

inline

Return the constant term \( f \) as described above.

Returns: The constant term \( f \)

virtual bool gtsam::HessianFactor::empty ( ) const

inlinevirtual

Check if the factor is empty.

TODO: How should this be defined?

Implements gtsam::GaussianFactor.

double gtsam::HessianFactor::error ( const VectorValues & c ) const

virtual

Evaluate the factor error f(x), see above.

Implements gtsam::GaussianFactor.

virtual DenseIndex gtsam::HessianFactor::getDim ( const_iterator variable ) const

inlinevirtual

0.5*[x -1]'H[x -1] (also see constructor documentation)

Return the dimension of the variable pointed to by the given key iterator todo: Remove this in favor of keeping track of dimensions with variables?

Parameters

variable An iterator pointing to the slot in this factor. You can use, for example, begin() + 2 to get the 3rd variable in this factor.

Implements gtsam::GaussianFactor.

constBlock gtsam::HessianFactor::info	(	const_iterator	j1,
		const_iterator	j2
	)		const

inline

Return a view of the block at (j1,j2) of the upper-triangular part of the information matrix \( H \), no data is copied.

See HessianFactor class documentation above to explain that only the upper-triangular part of the information matrix is stored and returned by this function.

Parameters

j1	Which block row to get, as an iterator pointing to the slot in this factor. You can use, for example, begin() + 2 to get the 3rd variable in this factor.
j2	Which block column to get, as an iterator pointing to the slot in this factor. You can use, for example, begin() + 2 to get the 3rd variable in this factor.

Returns: A view of the requested block, not a copy.

Block gtsam::HessianFactor::info	(	iterator	j1,
		iterator	j2
	)

inline

Return a view of the block at (j1,j2) of the upper-triangular part of the information matrix \( H \), no data is copied.

See HessianFactor class documentation above to explain that only the upper-triangular part of the information matrix is stored and returned by this function.

Parameters

j1	Which block row to get, as an iterator pointing to the slot in this factor. You can use, for example, begin() + 2 to get the 3rd variable in this factor.
j2	Which block column to get, as an iterator pointing to the slot in this factor. You can use, for example, begin() + 2 to get the 3rd variable in this factor.

Returns: A view of the requested block, not a copy.

SymmetricBlockMatrix::constBlock gtsam::HessianFactor::info ( ) const

inline

Return the upper-triangular part of the full augmented information matrix, as described above.

See HessianFactor class documentation above to explain that only the upper-triangular part of the information matrix is stored and returned by this function.

SymmetricBlockMatrix::Block gtsam::HessianFactor::info ( )

inline

Return the upper-triangular part of the full augmented information matrix, as described above.

See HessianFactor class documentation above to explain that only the upper-triangular part of the information matrix is stored and returned by this function.

std::pair< Matrix, Vector > gtsam::HessianFactor::jacobian ( ) const

virtual

Return (dense) matrix associated with factor.

Parameters

ordering	of variables needed for matrix column order
set	weight to true to bake in the weights

Implements gtsam::GaussianFactor.

constBlock::OffDiagonal::ColXpr gtsam::HessianFactor::linearTerm ( const_iterator j ) const

inline

Return the part of linear term \( g \) as described above corresponding to the requested variable.

Parameters

j	Which block row to get, as an iterator pointing to the slot in this factor. You can use, for example, begin() + 2 to get the 3rd variable in this factor.

Returns: The linear term \( g \)

Block::OffDiagonal::ColXpr gtsam::HessianFactor::linearTerm ( iterator j )

inline

Return the part of linear term \( g \) as described above corresponding to the requested variable.

Parameters

j	Which block row to get, as an iterator pointing to the slot in this factor. You can use, for example, begin() + 2 to get the 3rd variable in this factor.

Returns: The linear term \( g \)

constBlock::OffDiagonal::ColXpr gtsam::HessianFactor::linearTerm ( ) const

inline

Return the complete linear term \( g \) as described above.

Returns: The linear term \( g \)

Block::OffDiagonal::ColXpr gtsam::HessianFactor::linearTerm ( )

inline

Return the complete linear term \( g \) as described above.

Returns: The linear term \( g \)

const SymmetricBlockMatrix& gtsam::HessianFactor::matrixObject ( ) const

inline

Return the full augmented Hessian matrix of this factor as a SymmetricBlockMatrix object.

GaussianFactor::shared_ptr gtsam::HessianFactor::negate ( ) const

virtual

Construct the corresponding anti-factor to negate information stored stored in this factor.

Returns: a HessianFactor with negated Hessian matrices

Implements gtsam::GaussianFactor.

size_t gtsam::HessianFactor::rows ( ) const

inline

Return the number of columns and rows of the Hessian matrix, including the information vector.

void gtsam::HessianFactor::updateATA	(	const JacobianFactor &	update,
		const Scatter &	scatter
	)

Update the factor by adding the information from the JacobianFactor (used internally during elimination).

Parameters

update	The JacobianFactor containing the new information to add
scatter	A mapping from variable index to slot index in this HessianFactor

void gtsam::HessianFactor::updateATA	(	const HessianFactor &	update,
		const Scatter &	scatter
	)

Update the factor by adding the information from the HessianFactor (used internally during elimination).

Parameters

update	The HessianFactor containing the new information to add
scatter	A mapping from variable index to slot index in this HessianFactor

The documentation for this class was generated from the following files:

/home/cbeall3/git/gtsamB/gtsam/linear/HessianFactor.h
/home/cbeall3/git/gtsamB/gtsam/linear/HessianFactor-inl.h
/home/cbeall3/git/gtsamB/gtsam/linear/HessianFactor.cpp

Detailed Description

Public Member Functions

Public Types

Protected Attributes

Friends

Constructor & Destructor Documentation

Member Function Documentation